Download Astronomy, Astrophysics, and Cosmology

Astronomy, Astrophysics, and Cosmology
Luis A. Anchordoqui
Department of Physics and Astronomy
Lehman College, City University of New York
Lesson II
February 16, 2016
arXiv:0706.1988
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Table of Contents
1
Doppler Effect
2
Stellar Evolution
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Doppler Effect
Observational evidence that stars move
at speeds ranging up to a few hundred kilometers per second
E.g. + relatively fast moving Barnard’s star @ D ∼ 56 × 1012 km
moves across line of sight @ v ∼ 89 km/s
Consequence + proper motion shifts ∼ 0.0029◦ /yr
HST has measured proper motions as low as about 1 marcs/yr
In radio + VLBA relative motions
can be measured to an accuracy of ∼ 0.2 marcs/yr
Apparent position in sky of more distant stars changes so slowly
that proper motion can’t be detected
with even most patient observation
However + rate of approach or recession of star in line of sight
can be measured much more accurately
than its ⊥ motion to line of sight
Technique uses familiar property of any sort of wave motion
Doppler effect
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Doppler Effect
Barnard’s star
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Doppler Effect
When we observe sound or light wave from source at rest
time between arrival wave crests at our instruments
is same as time between crests as they leave source
However + if source is moving away from us
time between arrivals of successive wave crests
increases over time between departures from source
because each crest has little farther to go on its journey to us
than crest before
Time between crests + wavelength divided by wave speed
so a wave sent out by a source moving away from us
will appear to have longer wavelength than source @ rest
Likewise + if source is moving toward us
time between arrivals of wave crests is decreased
because each successive crest has shorter distance to go
and waves appear to have shorter wavelength
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Doppler Effect
redshift and blueshift
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Doppler Effect
Weinberg’s analogy
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Doppler Effect
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Doppler Effect
G Consider two inertial frames S and S0 moving with relative velocity v
G Assume star @ rest in S0 emits light @ (ν0 , θ0 ) with respect to O0
G Momentum 4-vector for photon as seen in S
hν
hν
hν
µ
p =
, − cos θ, − sin θ, 0
(1)
c
c
c
G Momentum 4-vector for photon as seen in S0
hν0
hν0
hν0
µ
p0 =
,−
cos θ0 , −
sin θ0 , 0
c
c
c
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(2)
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Doppler Effect
G Apply inverse LT to get 4-momentum relation from S0 → S
hν0
hν0
hν
= γ
+β −
cos θ0
c
c
c
hν
hν0
hν0
− cos θ = γ −
cos θ0 + β
c
c
c
hν0
hν
sin θ =
sin θ0
c
c
(3)
G First expression gives relativistic Doppler formula
ν = ν0 γ(1 − β cos θ0 )
(4)
G For observational astronomy (4) is not useful
because (ν0 , θ0 ) refer to the star’s frame not that of observer
G Apply instead direct LT S → S0 to photon energy
ν0 = γν(1 + β cos θ )
(5)
which gives ν0 in terms of quantities measured by observer
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Doppler Effect
Special cases
θ0 = 0 + source moving away from the observer
q
ν = ν0 (1 − β)/(1 + β)
(6)
Non-relativistic limit + ν = ν0 (1 − β)
θ0 = π + source moving towards observer
q
ν = ν0 (1 + β)/(1 − β)
(7)
θ0 = π/2 + transverse Doppler effect
ν = ν0 γ
(8)
2nd order relativistic effect
arising from dilation of time in moving frame
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Doppler Effect
Search for exoplanets
G As planet orbits star + star has its own orbit around CM system
G Radial velocity method: Variations in star’s radial velocity detected
from displacements in star’s spectral lines due to Doppler effect
G Photometric method: If planet crosses in front of its parent star’s disk
observed brightness drops by small amount
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Stellar Evolution
Stars appear unchanging
Night after night heavens reveal no significant variations
On human time scales + majority of stars change very little
We cannot follow any but tiniest part of star life cycle
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Stellar Evolution
Star formation
Stars are born when gaseous clouds (mostly hydrogen)
contract due to pull of gravity
Huge gas cloud fragments into numerous contracting masses
Each mass is centered in area
where density is only slightly greater than @ nearby points
Once such “globules” formed
gravity would cause each to contract in towards its CM
As particles of such protostar accelerate inward
their kinetic energy increases
When kinetic energy is sufficiently high
Coulomb repulsion + not strong enough to keep 1 H nuclei appart
and nuclear fussion can take place
In star like our Sun
“burning” of 1 H occurs when 4p fuse to form 2 He nucleus
with release of: γ, e+ , νe
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Stellar Evolution
M16 a.k.a. Eagle Nebula located ≈ 7, 000 ly away
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Stellar Evolution
HST’s pillars of creation
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Stellar Evolution
Sun’s energy output
pp cycle due to following sequence of fusion reactions:
1
1
1H +1H
→21H + 2 e+ + 2 νe
(0.42 MeV)
(9)
1
2
1H +1H
→32He + γ
(5.49 MeV)
(10)
(12.86 MeV)
(11)
3
3
2 He +2He
→42He +11H +11H
Released energy ã mass difference between initial & final states
â carried off by outgoing particles
4 11H →42He + 2e+ + 2νe + 2γ
(12)
Net effect
Takes 2 of each of first 2 reactions to produce two 32 He
Total energy released for net reaction + 24.7 MeV
e+ quickly annihilates with e− to produce 2me c2 = 1.02 MeV
so total energy released + 26.7 MeV
Deuterium formation has very low probability
infrequency of reaction limits rate at which Sun produces energy
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Stellar Evolution
In more massive stars...
Energy output comes from the carbon (or CNO) cycle
CNO cycle comprises following sequence of reactions:
12
1
6C +1H
→137N + γ
(13)
→136C + e+ + ν
(14)
13
1
6C +1H
→147N + γ
(15)
14
1
7N +1H
→158O + γ
(16)
13
7N
15
8O
15
1
7N +1H
→157N + e+
+ν
(17)
→126C +42He
(18)
No carbon is consumed in this cycle (see first and last equations)
Net effect is the same as the pp cycle
Theory of the pp and CNO cyles first worked out by Bethe in 1939
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Stellar Evolution
Nuclear Burning
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Stellar Evolution
Fusion reactions take place in star core + T ∼ 107 K
Surface temperature is much lower + O(few thousand K)
Tremendous release of energy in these fusion reactions
produces outward pressure to halt inward gravitational contraction
protostar (now really a young star) stabilizes in main sequence
Stellar structure on main sequence + described by
spherically symmetric system in hydrostatic equilibrium
This requires that rotation, convection, magnetic fields,
and other effects that break rotational symmetry
have only a minor influence on the star
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Stellar Evolution
G M(r ) + mass enclosed inside sphere with radius r and density ρ(r )
M (r ) = 4π
Z r
0
2
dr 0 r 0 ρ(r 0 ) ⇒
dM (r )
= 4πr2 ρ(r )
dr
(19)
G Gravitational acceleration produced by M (r ) is
g (r ) = −
GM(r )
r2
(20)
G If star is in equilibrium + acceleration balanced by
pressure gradient from center of star to its surface
G Since P = F/A + pressure change along distance dr yields
dF = dAP − ( P + dP)dA
(r )dAdr
= − dAdP
| {z } = − ρ
| {z }
force
mass
a (r )
|{z}
(21)
acceleration
G For increasing r + gradient dP < 0
and resulting dF is positive and directed outward
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Stellar Evolution
G Hydrostatic equilibrium + g(r ) = − a(r )
GM(r ) ρ(r )
dP
= ρ (r ) g (r ) = −
(22)
dr
r2
G If the pressure gradient and gravity do not balance each other
layer at position r is accelerated
a (r ) =
GM(r )
1 dP
+
r2
ρ(r ) dr
(23)
G In general + equation of state P = P(ρ, T, Yi )
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Stellar Evolution
G Estimate of central pressure + Pc = P(0)
integrate (22) using (19) and obtain with P( R) ≈ 0
Pc =
Z R
dP
0
dr
dr = G
Z M
dM
0
M
,
4πr4
(24)
G Pc lower limit + replace r by stellar radius R ≥ r
Z M
M
4
4πr
0
Z M
M
M2
> G
dM
=
4πR4
8πR4
0
Pc = G
dM
(25)
G Inserting values for Sun
M2
Pc >
= 4 × 108 bar
8πR4
M
M
2 R
R
4
(26)
G Integrating hydrostatic equation using “solar standard model”
Pc = 2.48 × 1011 bar + factor 500 larger
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Stellar Evolution
to be continued . . .
Literature: Longair: §8, 10, 12 & 13
HEA 09/ p
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